Error bounds for the (KdV)/(KP-I) and (gKdV)/(gKP-I) asymptotic regime for nonlinear Schrödinger type equations
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 6, pp. 1175-1230.

We consider the (KdV)/(KP-I) asymptotic regime for the nonlinear Schrödinger equation with a general nonlinearity. In a previous work, we have proved the convergence to the Korteweg–de Vries equation (in dimension 1) and to the Kadomtsev–Petviashvili equation (in higher dimensions) by a compactness argument. We propose a weakly transverse Boussinesq type system formally equivalent to the (KdV)/(KP-I) equation in the spirit of the work of Lannes and Saut, and then prove a comparison result with quantitative error estimates. For either suitable nonlinearities for (NLS) either a Landau–Lifshitz type equation, we derive a (mKdV)/(mKP-I) equation involving cubic nonlinearity. We then give a partial result justifying this asymptotic limit.

DOI: 10.1016/j.anihpc.2013.08.007
Classification: 35Q55,  35Q53
Keywords: Nonlinear Schrödinger equation, Gross–Pitaevskii equation, Landau–Lifshitz equation, (Generalized) Korteweg–de Vries equation, (Generalized) Kadomtsev–Petviashvili equation, Weakly transverse Boussinesq system
@article{AIHPC_2014__31_6_1175_0,
     author = {Chiron, D.},
     title = {Error bounds for the {(KdV)/(KP-I)} and {(gKdV)/(gKP-I)} asymptotic regime for nonlinear {Schr\"odinger} type equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1175--1230},
     publisher = {Elsevier},
     volume = {31},
     number = {6},
     year = {2014},
     doi = {10.1016/j.anihpc.2013.08.007},
     zbl = {1307.35274},
     mrnumber = {3280065},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2013.08.007/}
}
TY  - JOUR
AU  - Chiron, D.
TI  - Error bounds for the (KdV)/(KP-I) and (gKdV)/(gKP-I) asymptotic regime for nonlinear Schrödinger type equations
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2014
DA  - 2014///
SP  - 1175
EP  - 1230
VL  - 31
IS  - 6
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2013.08.007/
UR  - https://zbmath.org/?q=an%3A1307.35274
UR  - https://www.ams.org/mathscinet-getitem?mr=3280065
UR  - https://doi.org/10.1016/j.anihpc.2013.08.007
DO  - 10.1016/j.anihpc.2013.08.007
LA  - en
ID  - AIHPC_2014__31_6_1175_0
ER  - 
%0 Journal Article
%A Chiron, D.
%T Error bounds for the (KdV)/(KP-I) and (gKdV)/(gKP-I) asymptotic regime for nonlinear Schrödinger type equations
%J Annales de l'I.H.P. Analyse non linéaire
%D 2014
%P 1175-1230
%V 31
%N 6
%I Elsevier
%U https://doi.org/10.1016/j.anihpc.2013.08.007
%R 10.1016/j.anihpc.2013.08.007
%G en
%F AIHPC_2014__31_6_1175_0
Chiron, D. Error bounds for the (KdV)/(KP-I) and (gKdV)/(gKP-I) asymptotic regime for nonlinear Schrödinger type equations. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 6, pp. 1175-1230. doi : 10.1016/j.anihpc.2013.08.007. http://archive.numdam.org/articles/10.1016/j.anihpc.2013.08.007/

[1] M. Abid, C. Huepe, S. Metens, C. Nore, C.T. Pham, L.S. Tuckerman, M.E. Brachet, Gross–Pitaevskii dynamics of Bose–Einstein condensates and superfluid turbulence, Fluid Dyn. Res. 33 no. 5–6 (2003), 509 -544 | MR | Zbl

[2] N. Akhmediev, A. Ankiewicz, R. Grimshaw, Hamiltonian-versus-energy diagrams in soliton theory, Phys. Rev. E 59 no. 5 (1999), 6088 -6096 | MR

[3] T. Alazard, R. Carles, Supercritical geometric optics for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal. 94 no. 1 (2009), 315 -347 | MR | Zbl

[4] V. Banica, E. Miot, Global existence and collisions for certain configurations of nearly parallel vortex filaments, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 29 (2012), 813 -832 | Numdam | MR | Zbl

[5] I. Barashenkov, E. Panova, Stability and evolution of the quiescent and travelling solitonic bubbles, Physica D 69 no. 1–2 (1993), 114 -134 | Zbl

[6] W. Ben Youssef, T. Colin, Rigorous derivation of Korteweg–de Vries-type systems from a general class of nonlinear hyperbolic systems, M2AN Math. Model. Numer. Anal. 34 no. 4 (2000), 873 -911 | EuDML | Numdam | MR | Zbl

[7] W. Ben Youssef, D. Lannes, The long wave limit for a general class of 2D quasilinear hyperbolic problems, Commun. Partial Differ. Equ. 27 no. 5–6 (2002), 979 -1020 | Zbl

[8] S. Benzoni-Gavage, R. Danchin, S. Descombes, On the well-posedness of the Euler–Korteweg model in several space dimensions, Indiana Univ. Math. J. 56 (2007), 1499 -1579 | MR | Zbl

[9] N. Berloff, P. Roberts, Motions in a Bose condensate: X. New results on stability of axisymmetric solitary waves of the Gross–Pitaevskii equation, J. Phys. A, Math. Gen. 37 (2004), 11333 -11351 | MR | Zbl

[10] F. Béthuel, R. Danchin, D. Smets, On the linear wave regime of the Gross–Pitaevskii equation, J. Anal. Math. 110 (2010), 297 -338 | MR | Zbl

[11] F. Béthuel, P. Gravejat, J.-C. Saut, On the KP-I transonic limit of two-dimensional Gross–Pitaevskii travelling waves, Dyn. Partial Differ. Equ. 5 no. 3 (2008), 241 -280 | MR | Zbl

[12] F. Béthuel, P. Gravejat, J.-C. Saut, Existence and properties of travelling waves for the Gross–Pitaevskii equation, Stationary and Time Dependent Gross–Pitaevskii Equations, Contemp. Math. vol. 473 , Amer. Math. Soc., Providence, RI (2008), 55 -103 | MR | Zbl

[13] F. Béthuel, P. Gravejat, J.-C. Saut, D. Smets, On the Korteweg–de Vries long-wave transonic approximation of the Gross–Pitaevskii equation I, Int. Math. Res. Not. 14 (2009), 2700 -2748 | MR | Zbl

[14] F. Béthuel, P. Gravejat, J.-C. Saut, D. Smets, On the Korteweg–de Vries long-wave approximation of the Gross–Pitaevskii equation II, Commun. Partial Differ. Equ. 35 no. 1 (2010), 113 -164 | MR | Zbl

[15] A. Capella, C. Melcher, F. Otto, Wave-type dynamics in ferromagnetic thin films and the motion of Néel walls, Nonlinearity 20 no. 11 (2007), 2519 -2537 | MR | Zbl

[16] D. Chiron, Travelling waves for the nonlinear Schrödinger equation with general nonlinearity in dimension one, Nonlinearity 25 (2012), 813 -850 | MR | Zbl

[17] D. Chiron, Three long-wave asymptotic regimes for the nonlinear Schrödinger Equation, M. Novaga, G. Orlandi (ed.), Singularities in Nonlinear Evolution Phenomena and Applications, CRM Series , Scuola Normale Superiore, Pisa (2009), 107 -138 | MR

[18] D. Chiron, M. Mariş, Rarefaction pulses for the nonlinear Schrödinger equation in the transonic limit, Commun. Math. Phys. (2013) | MR | Zbl

[19] D. Chiron, F. Rousset, Geometric optics and boundary layers for nonlinear Schrödinger equations, Commun. Math. Phys. 288 no. 2 (2009), 503 -546 | MR | Zbl

[20] D. Chiron, F. Rousset, The KdV/KP-I limit of the nonlinear Schrödinger equation, SIAM J. Math. Anal. 42 no. 1 (2010), 64 -96 | MR | Zbl

[21] D. Chiron, C. Scheid, Travelling waves for the nonlinear Schrödinger equation with general nonlinearity in dimension two, preprint. | MR

[22] T. Colin, D. Lannes, Justification of and long-wave correction to Davey–Stewartson systems from quadratic hyperbolic systems, Discrete Contin. Dyn. Syst. 11 no. 1 (2004), 83 -100 | MR | Zbl

[23] T. Colin, A. Soyeur, Some singular limits for evolutionary Ginzburg–Landau equations, Asymptot. Anal. 13 (1996), 361 -372 | MR | Zbl

[24] A. De Bouard, J.-C. Saut, Solitary waves of generalized Kadomtsev–Petviashvili equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14 no. 2 (1997), 211 -236 | EuDML | Numdam | MR | Zbl

[25] A. de Laire, Travelling waves for the Landau–Lifshitz equation: nonexistence of small energy solutions and asymptotic behaviour at infinity, preprint.

[26] W. Ding, Y. Wang, Local Schrödinger flow into Kähler manifolds, Sci. China Ser. A 44 no. 11 (2001), 1446 -1464 | MR | Zbl

[27] C. Gallo, Schrödinger group on Zhidkov spaces, Adv. Differ. Equ. 9 no. 5–6 (2004), 509 -538 | MR | Zbl

[28] C. Gallo, The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at infinity, Commun. Partial Differ. Equ. 33 no. 4–6 (2008), 729 -771 | MR | Zbl

[29] P. Gérard, Remarques sur l'analyse semi-classique de l'équation de Schrödinger non linéaire, Séminaire sur les Equations aux Dérivées Partielles, Exp. No. XIII, Ecole Polytechnique, Palaiseau (1992)(1993) | EuDML | MR

[30] P. Germain, F. Rousset, Long wave limits for Schrödinger maps, preprint.

[31] E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time, Proc. Am. Math. Soc. 126 no. 2 (1998), 523 -530 | MR | Zbl

[32] R. Iório, W. Nunes, On equations of KP-type, Proc. R. Soc. Edinb. A 128 (1998), 725 -743 | MR | Zbl

[33] Y. Kivshar, D. Anderson, M. Lisak, Modulational instabilities and dark solitons in a generalized nonlinear Schrödinger equation, Phys. Scr. 47 (1993), 679 -681

[34] Y.S. Kivshar, B. Luther-Davies, Dark optical solitons: physics and applications, Phys. Rep. 298 (1998), 81 -197

[35] Y. Kivshar, D. Pelinovsky, Self-focusing and transverse instabilities of solitary waves, Phys. Rep. 331 (2000), 117 -195 | MR

[36] E.B. Kolomeisky, T.J. Newman, J.P. Straley, X. Qi, Low-dimensional bose liquids: beyond the Gross–Pitaevskii approximation, Phys. Rev. Lett. 85 (2000), 1146 -1149

[37] S. Komineas, N. Papanicolaou, Topology and dynamics in ferromagnetic media, Physica D 99 no. 1 (1996), 81 -107 | MR | Zbl

[38] S. Komineas, N. Papanicolaou, Vortex dynamics in two-dimensional antiferromagnets, Nonlinearity 11 no. 2 (1998), 265 -290 | MR | Zbl

[39] L. Landau, E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion 8 (1935), 153 -169 | Zbl

[40] D. Lannes, Consistency of the KP approximation, Discrete Contin. Dyn. Syst. no. suppl. (2003), 517 -525 | MR | Zbl

[41] D. Lannes, Secular growth estimates for hyperbolic systems, J. Differ. Equ. 190 no. 2 (2003), 466 -503 | MR | Zbl

[42] D. Lannes, J.-C. Saut, Weakly transverse Boussinesq systems and the Kadomtsev–Petviashvili approximation, Nonlinearity 19 (2006), 2853 -2875 | MR | Zbl

[43] H. Leblond, KP lumps in ferromagnets: a three-dimensional KdV-Burgers model, J. Phys. A 35 no. 47 (2002), 10149 -10161 | MR | Zbl

[44] K. Nakamura, T. Sasada, Quantum kink in the continuous one-dimensional Heisenberg ferromagnet with easy plane: a picture of the antiferromagnetic magnon, J. Phys. C, Solid State Phys. 15 (1982), L1013 -L1017

[45] N. Papanicolaou, P. Spathis, Semitopological solitons in planar ferromagnets, Nonlinearity 12 no. 2 (1999), 285 -302 | MR | Zbl

[46] P. Roberts, N. Berloff, Nonlinear Schrödinger equation as a model of superfluid helium, C.F. Barenghi, R.J. Donnelly, W.F. Vinen (ed.), Quantized Vortex Dynamics and Superfluid Turbulence, Lect. Notes Phys. vol. 571 , Springer-Verlag (2001)

[47] J.-C. Saut, L. Xu, The Cauchy problem on large time for surface waves Boussinesq systems, J. Math. Pures Appl. (9) 97 no. 6 (2012), 635 -662 | MR | Zbl

[48] S. Schochet, Asymptotics for symmetric hyperbolic systems with a large parameter, J. Differ. Equ. 75 no. 1 (1988), 1 -27 | MR | Zbl

[49] J. Shatah, C. Zeng, Schrödinger maps and anti-ferromagnetic chains, Commun. Math. Phys. 262 (2006), 299 -315 | MR | Zbl

[50] P.-L. Sulem, C. Sulem, C. Bardos, On the continuous limit for a system of classical spins, Commun. Math. Phys. 107 no. 3 (1986), 431 -454 | MR | Zbl

[51] M. Taylor, Partial Differential Equations (III), Appl. Math. Sci. vol. 117 , Springer-Verlag, New York (1997) | MR

[52] T. Tsuzuki, Nonlinear waves in the Pitaevskii–Gross equation, J. Low Temp. Phys. 4 no. 4 (1971), 441 -457

[53] S. Ukaï, Local solutions to the Kadomtsev–Petviashvili equation, J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 36 (1989), 193 -209 | MR | Zbl

[54] L. Xu, Intermediate long wave systems for internal waves, Nonlinearity 25 no. 3 (2012), 597 -640 | MR | Zbl

[55] V. Zakharov, A. Kuznetsov, Multi-scale expansion in the theory of systems integrable by the inverse scattering transform, Physica D 18 no. 1–3 (1986), 455 -463 | MR | Zbl

[56] P. Zhidkov, Korteweg–de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lect. Notes Math. vol. 1756 , Springer-Verlag (2001) | MR | Zbl

Cited by Sources: